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Anomalous $ω$-$Z$-$γ$ Vertex from Hidden Local Symmetry

We formulate the general form of omega-Z-gamma vertex in the framework based on the hidden local symmetry (HLS), which arises from the gauge invariant terms for intrinsic parity-odd (IP-odd) part of the effective action. Those terms are given as the homogeneous part of the general solution (having free parameters) to the Wess-Zumino (WZ) anomaly equation and hence are not determined by the anomaly, in sharp contrast to the Harvey-Hill-Hill (HHH) action where the relevant vertex is claimed to be uniquely determined by the anomaly. We show that, even in the framework that HHH was based on, the omega-Z-gamma vertex is actually not determined by the anomaly but by the homogeneous (anomaly-free) part of the general solution to the WZ anomaly equation having free parameters in the same way as in the HLS formulation: The HHH action is just a particular choice of the free parameters in the general solution. We further show that the omega-Z-gamma vertex related to the neutrino (nu) - nucleon (N) scattering cross section sigma(nu N -> nu N (N')gamma) is determined not by the anomaly but by the anomaly-free part of the general solution having free parameters. Nevertheless we find that the cross section sigma(nu N -> nu N(N')gamma) is related through the Ward-Takahashi identity to Gamma(omega -> pi^0 gamma) which has the same parameter-dependence as that of sigma(nu N -> nu N(N')gamma) and hence the ratio sigma(nu N -> nu N(N')gamma)/Gamma(omega -> pi^0 gamma) is fixed independently of these free parameters. Other set of the free parameters of the general solution can be fixed to make the best fit of the omega -> pi^0 l^+ l^- process, which substantially differs from the HHH action. This gives a prediction of the cross section sigma(nu N -> nu N (N')gamma^*(l^+ l^-)) to be tested at nu-N collision experiments in the future.